Question

How many ways can you roll a pair of dice and get an even product?

Answer 1

27

Explanation:

The definition of an even number is `2k` and for odd `2k+1`. We know that a dice has 6 sides and when there are 2 there is a possible of `6^2=36` outcomes. This corresponds to each side from die 1 with each of the 6 sides from die 2.

In this case we are interested in only even outcomes.

If die 1 is even and die 2 is even then it will be even because if `2m,2n` are even then `2m * 2n = 4mn =2(2mn)`.

If die 1 is even and die 2 is odd the the product is even because if x is even and y is odd then let `x=2m, y=2n+1` and `(2m)(2n+1)=4nm+2m = 2(2nm+m)` which is even. The same is true if die 1 is odd and die 2 is even.

So if die 1 is even then all the 6 sides from die 2 will be even. If die one is odd then all the even numbers of die 2 are even.

The last case is if both dice are odd. It turns out that this too is odd because `(2m+1) * (2n+1) = 4mn+2n+2m+1 = 2(2mn+m+n)+1`

So for 2,4,6 all 6 from two are even and for 1,3,5 only 2,4,6 are even thus `3×6 +3×3=27` even outcome